Applying a Simple Cylindrical Projection onto a Object
You can use an environment texture in order to project on the mesh surface.



The environment texture is generally used for scene environment lightning, but it performs an equirectangular projection that seems to meet our need here.

Take the object output from the texture coordinates node.

Plug it into a vector mapping.

And plug its result into the env texture node.

Now we just have to handle what needs to be rotated in the mapping (it's a guess): here a rotation of 180 around X and a -1 scale for Z seem to correspond perfectly to the mesh.

They are little distortions on the left part, but all craters have found the good place, it seems.

The texture from the link in the question is not included as too large to be uploaded with the blend file:



Note: 'equirectangular projection' is also called 'equidistant cylindrical projection'